solve x^4+2x^3-10x^2-14x-3=0

Question

solve the polynomial equation using synthetic division.

Answer 1

x^4+2x^3-10x^2-14x-3=0                                                                                                                       the rational roots theorem says that possible roots are +(factors of 3)/(factors of 1)(the coefficient of x^4 is 1)

let's try 3

 

3     1   2   -10   -14   -3

           3     15    15    3

      

       1   5     5      1     0

 

(x^4+2x^3-10x^2-14x-3)/(x-3)=x^3+5x^2+5x+1

 

using x^3+5x^2+5x+1

  

let's try -1

 

-1       1   5   5   1

              -1  -4  -1

          

          1   4   1    0

 

(x^3+5x^2+5x+1)/(x+1)=x^2+4x+1

so far we have (x-3)(x+1)(x^2+4x+1)=0

 

use the quadratic formula for x^2+4x+1

 

[-4+sqrt(16-4)]/2=[-4+sqrt(12)]/2=[-4+2sqrt(3)]/2=-2+sqrt(3)

 

x=3, -1,  -2+sqrt(3),  -2-sqrt(3)

 

 

 

 

          

 

     

      

 

       

         

 

Answer 2

Hey Caprice -- roots to try => (+/- factors of last term) / (+/- factors of leading term)

 

... x = -1,3 work => divide by (x+1)(x-3)

 

... (x^2 -2x -3) divisor yields => x^2 +4x +1 ... reframe (x^2 +4x +4) -3 => (x+2)^2 -3

 

... x+2 = +/- sqrt 3 ==> roots x= -1, 3, -2 +/- sqrt 3 ... Best regards :)

Answer 3

x = -(√3 +2), or x = -1, or x = √3 - 2, or x = 3

 

I apologize, Caprice. The answer above was returned by my TI-89 Titanium calculator.  I really don't know how to prove this answer via synthetic division.

 

Here is a link to a page with a synthetic division calculator: http://www.mathportal.org/calculators/polynomials-solvers/synthetic-division-calculator.php